On Universal and Fault-Tolerant Quantum Computing

On Universal and Fault-Tolerant Quantum Computing ∗ P. Oscar Boykin, Tal Mor, Matthew Pulver, Vwani Roychowdhury, and Farrokh Vatan

†

arXiv:quant-ph/9906054v1 16 Jun 1999

Electrical Engineering Department UCLA Los Angeles, CA 90095

Abstract

quirements for error-free operations are to have a set of gates that is both universal for quantum A novel universal and fault-tolerant basis (set computing (see [4] and references therein), and of gates) for quantum computation is described. that can operate in a noisy environment (i.e., Such a set is necessary to perform quantum fault-tolerant) [23, 25, 2, 18, 17]. computation in a realistic noisy environment. The new basis consists of two single-qubit gates A scheme to correct errors in quantum bits 1 (Hadamard and σz 4 ), and one double-qubit gate (qubits) was proposed by Shor [23] by adopt(Controlled-NOT). Since the set consisting of ing standard coding techniques and modifying Controlled-NOT and Hadamard gates is not uni- them to correct quantum mechanical errors inversal, the new basis achieves universality by in- duced by the environment. In such quantum cluding only one additional elementary (in the error-correction techniques, the two states of sense that it does not include angles that are each qubit are encoded using a string of qubits, irrational multiples of π) single-qubit gate, and so that the state of the qubit is kept in a prehence, is potentially the simplest universal basis specified two-dimensional subspace of the space that one can construct. We also provide an al- spanned by the string of qubits. We refer to ternative proof of universality for the only other this as the logical qubit. This is done in a way known class of universal and fault-tolerant basis that error in one or more (as permitted by the proposed in [25, 17]. code) physical qubits will not destroy the logical qubit. To avoid errors in the computation itself, Shor [25] suggested performing the com1 Introduction putations on the logical qubits (without first decoding them), and this type of computation is A new model of computation based on the laws known as fault-tolerant computation. of quantum mechanics has been shown to be superior to standard (classical) computation modThere are a number of requirements that a els [24, 15]. Potential realizations of such com- fault-tolerant quantum circuit must satisfy. To puting devices are currently under extensive re- prevent propagation of single-qubit errors to search [7, 20, 29, 10, 13, 12, 16, 30], and the other qubits in the same code word, one requiretheory of using them in a realistic noisy envi- ment of fault-tolerant computation is to disalronment is still developing. Two of the main re- low operations between any two qubits from the same codeword. This constraint imposes significant restrictions on both the types of unitary operations that can be performed on the encoded logical qubits, and the quantum errorcorrecting codes that can be used to encode the logical qubits. For example, if a “double-even”

∗

This work was supported in part by grants from the Revolutionary Computing group at JPL (contract #961360), and from the DARPA Ultra program (subcontract from Purdue University #530–1415–01). † E–mail addresses of the authors are, respectively: {boykin, talmo, pulver, vwani, vatan}@ee.ucla.edu.

1

CSS code1 (e.g., the ((7, 1, 3)) quantum code described in [6, 28]) is used then one can show that the following unitary operations can be faulttolerantly implemented:   1 0 0 0 1  0 1 0 0   H, σz2 , ∧1 (σx ) =  (1)  0 0 0 1  0 0 1 0

basis, however, was not included. Later Kitaev [17] proved the universality of a basis, compris1 ing the set {∧1 (σz 2 ), H} (see Section 5.1), that is “equivalent” to Shor’s basis, i.e., the gates in the new basis can be exactly realized using gates in Shor’s basis and vice-versa. This result provides the first published proof of the universality of a fault tolerant basis. A number of other researchers have proposed fault-tolerant bases that are equivalent to Shor’s basis. Knill, Laflamme, and Zurek [19] con1 1 sidered the bases { σz 2 , Λ1 (σz 2 ), Λ1 (σx ) } and 1 1 { H, σz 2 , Λ1 (σz 2 ), Λ1 (σx ) }. The universality of these bases follows from the fact that gates in Shor’s basis can be simulated by small simple circuits over these new bases. Hence, while novel fault-tolerant realizations of the relevant gates in these bases were proposed, no new proofs of universality was provided. The same authors later [18] studied a model in which the following set of 1 gates { H, σz 2 , Λ1 (σx ) } and the prepared state cos(π/8)|0iL +sin(π/8)|1iL 2 are made available. Again, the universality of this model follows from the fact that it can realize the gate Λ1 (H), and consequently the Toffoli gate. We also note that Aharonov and Ben-Or [2] considered quantum systems with basic units that have p > 2 states (referred to as qupits). They proposed a class of quantum codes, called polynomial codes, for such systems consisting of qupits. They defined a basis for the polynomial codes and proved that it is universal. However their proof makes explicit use of qupits with more than two states, and hence does not directly apply to the case studied in this paper, where all operations are done on qubits only.

1 2

where H and σz are defined in the next section, and ∧k (U ) denotes the controlled–U operation with k control bits (see [4]). So far, these operations are the only ones that have been shown to be ”directly” fault-tolerant (in the sense that no measurements and/or preparations of special states are required) operations. It is well known, however, that the group generated by the above operations (also referred to as the normalizer group) is not universal for quantum computation. This leads to the interesting problem of determining a basis that is both universal and can be implemented fault-tolerantly. There are several well-established results on the universality of quantum bases [1, 3, 4, 5, 9]. Proofs of universality of these bases rest primarily on the fact that they include at least one “non-elementary” gate, i.e., a gate that performs a rotation on single qubits by an irrational multiple of π. A direct fault-tolerant realization of such a gate, however, is not possible; this property makes all the well-known universal bases inappropriate for practical and noisy quantum computation. The search for universal and fault-tolerant bases has led to a novel basis as proposed in the seminal work of Shor [25]. It includes the Toffoli gate in addition to the above-mentioned generators of the normalizer group; hence, the basis can be represented by the following set 1 { H, σz 2 , Λ2 (σx ) }. A fault-tolerant realization of the Toffoli gate (involving only the generators of the normalizer group, preparation of a special state, and appropriate measurements) has been shown in [25]; a proof of universality of this

In this paper, we prove the existence of a novel basis for quantum computation that lends itself to an elegant proof (based solely on the geometry of real rotations in three dimensions) of universality and in which all the gates can be easily realized in a fault-tolerant manner. In fact, we show that the inclusion of only one additional

1

Double even codes are punctured binary codes in which the number of ones in any codeword is a multiple of 4 [25].

2 |0iL and |1iL refer to the states of the logical/encoded qubits.

2

single-qubit operation in the set in (1), namely,   1 1 0 4 π σz ≡ 0 ei 4

where n ˆ is a normalized real three-dimensional vector. From commutation relations one can show that (ˆ n · ~σ )2 = I, and using this fact, exponentiation of these Pauli matrices can be easily expanded as follows:

leads to a universal and fault-tolerant basis for 1 2 quantum computation. Note that σz is not reeiφˆn·~σ = cos φ I + i sin φ(ˆ n · ~σ ) quired anymore. Thus, our basis consists of the where I is the 2×2 identity matrix. It should be following three gates noted that for every element, U in SU(2) there 1 4 ˆ U , such that [26] H, σz , ∧1 (σx ) . (2) exist φU and n U = eiφU nˆ U ·~σ .

Proving the universality of Shor’s basis (see [17]) seems to be a more involved process than proving the universality of the set of gates we suggest here. Moreover, we outline a general method for fault tolerant realizations of a certain class of unitary operations; the fault-tolerant realiza1 tions of both the σz 4 and the Toffoli gates are shown to be special cases of this general formulation. The first part of this paper is devoted to the proof of universality, followed by a discussion on 1 the fault-tolerant realization of the σz 4 gate, and finally an alternate proof for the universality of Shor’s basis. We also show in Appendix A that the new basis proposed in this paper is not equivalent to Shor’s basis.

2

Next, to motivate our proof, we note the connection between real rotations in three dimensions (i.e., elements of SO(3)) and the group we are concerned with, SU(2). Note that Euler decompositions provide a way to represent a general rotation by an angle 2φ about an axis n ˆ, Rnˆ (2φ), by a product of rotations about two orthogonal axes. That is, Rnˆ (2φ) = Rzˆ(2α)Ryˆ (2β)Rzˆ(2γ).

There is a local isomorphism between SO(3) and SU(2). For the same parameters in (3), the following equation is also true: eiφˆn·~σ = eiασz eiβσy eiγσz .

Definitions and Identities

σy ≡ iσx σz = H≡

√1 (σx 2

0 −i i 0



;

√1 2



1 1 1 −1

+ σz ) =



(4)

Thus, just as any rotation can be thought of as three rotations about two axes, any element of SU(2) can be thought of as a product of three matrices, specifically, powers of exponentials of Pauli matrices. In the following section we will show that using the operations in our basis (2), we can approximate any “rotation” about two specific orthogonal axes, and then by Euler decomposition, we will show how all elements of SU(2) can be approximated. Lastly, we introduce a set of notations involving real powers of the Pauli matrices that prove to be very useful. For example, there is an obvious way to raise σz to a real power:   1 0 α . σz = 0 eiπα

Define the usual Pauli matrices σi , and Hadamard operator H.     0 1 1 0 σx ≡ ; σz ≡ ; 1 0 0 −1 

(3)

.

First, we review some properties of matrices in SU(2). Note that all traceless and Hermitian 2×2 unitary matrices can be represented as follows:

These matrices form an interesting family for quantum computation, and are generally used

n ˆ · ~σ ≡ nx σx + ny σy + nz σz , 3

to put a relative phase between |0i and |1i. For ~n2 are orthogonal. (These would need to be nor1 example, σz2n , for integer values of n, are used malized when used in exponentiation.) The number ei2πλ is a root of the irreducible in Shor’s Factorization algorithm [24]. Based on the definition of σ α , we use similarity transfor- monic polynomial z

mation to make the following identities: −1

1

1

−1

σxα = Hσzα H, σyα = σz2 σxα σz 2 , H α ≡ σy4 σzα σy 4 . Note that we can also equivalently write πα πα ei 2 e−i 2 σj .

3

σjα

A Proof of Universality

1 x4 + x3 + x2 + x + 1 4

which is not cyclotomic (since not all coefficients = are integers), and thus λ is an irrational number (see Appendix B Theorem B.1). Since λ is irrational, it can be used to approximate any phase factor eiφ : einλπ ≈ eiφ

The proof of universality of our basis will be broken down into two steps. In the first step we for some n ∈ N. 1 Now we at least have two dense subsets of show that H and σz4 form a dense set in SU(2); iαˆ n1 ·~ σ and eiβ n ˆ 2 ·~ σ: i.e. for any element of SU(2) and desired degree SU(2). Namely, e of precision, there exists a finite product of H 1

and σz4 that approximates it to this desired degree of precision. Next we observe that for universal quantum computation all that is needed is ∧1 (σx ) and SU(2) [4]. For proving density in SU(2) using our basis, we first show that we can construct elements in our basis which correspond to rotations by angles that are irrational multiples of π in SO(3) about two orthogonal axes. Once we have these irrational rotations about two orthogonal axes, then the density in SU(2) follows simply from the local isomorphism between SU(2) and SO(3) discussed in the previous section. Let −1

1

−1

1

(mod 2π)

∃j ∈ N such that λπk ≈ β

(mod 2π)

iλπˆ n1 ·~ σ j

hence, (e

) ≈ eiαˆn1 ·~σ .

iλπˆ n2 ·~ σ k

hence, (e

) ≈ eiβ nˆ 2 ·~σ .

(10)

(11)

Fortunately, this is all that is needed. Since, ~n1 and ~n2 are orthogonal, we can write any element of SU(2) in the following form [26]: eiφˆn·~σ = (eiαˆn1 ·~σ )(eiβ nˆ 2 ·~σ )(eiγ nˆ 1 ·~σ ) .

(12)

Clearly, the representation in (12) is analogous to Euler rotations about three orthogonal vectors. (5) Expansion of (12) gives:

1

eiλπˆn1 ·~σ ≡ σz 4 σx4 and

∃j ∈ N such that λπj ≈ α

1

eiλπˆn2 ·~σ ≡ H − 2 σz 4 σx4 H 2 .

(6)

cos φ = cos β cos(γ + α) n ˆ sin φ = n ˆ 1 cos β sin(γ + α)

Given (5,6) one can calculate ~n1 , ~n2 , and λ: π 1 1 cos λπ = cos2 = (1 + √ ) 8 2 2 √ π zˆ − x ˆ ~n1 = ( 2 cot ) √ + yˆ 8 2 √ π zˆ − x ˆ ~n2 = ( 2 cot )ˆ y− √ 8 2

(13)

+n ˆ 2 sin β cos(γ − α)

(14)

+n ˆ1 × n ˆ 2 sin β sin(γ − α)

(7)

For any element of SU(2) equations (13,14) can (8) , be inverted to find α, β and γ. Since ∧1 (σx ) and SU(2) form a universal basis for quantum (9) computation [4], it completes the proof of universality of our basis. where x ˆ, yˆ, and zˆ are the unit vectors along the There is no guarantee that, in general, it is respective axes. One can easily verify that ~n1 , possible to efficiently approximate an arbitrary

4

1

|ψiL

σz2

|ψ ′ iL

1

|ψiL

|ψ ′ iL

σz4

= |φ0 iL

M 1

Figure 1: Implementing σz4 fault-tolerantly. phase eiφ by repeated applications of the available phase eiπλ . But using an argument similar to the one presented in [1], we can show that for the given λ (as defined in (7)), for any given ε > 0, with only poly( 1ε ) iterations of eiπλ we can get eiφ within ε. However, since our basis is already proven to be universal, one can make use of an even better result. As it is shown by Kitaev [17] and Solovay and Yao [27], every universal quantum basis B is efficient, in the sense that any unitary operation in U(2m ), for constant m, can be approximated within ε by a circuit of size poly-log( 1ε ) over the basis B.

ation, as demonstrated in the following: π

|ψi ⊗ |φ0 i

=

(α|0i + β|1i) ⊗

∧1 (σx )

−→

1

π

|0i+ei 4 |1i √ 2

(α|0i + ei 4 β|1i) ⊗

|0i √ 2 π

π

+(α|0i + e−i 4 β|1i) ⊗ ei 4

= σz4 |ψi ⊗

|0i √ 2

−1

π

+ σz 4 |ψi ⊗ ei 4

|1i √ 2

|1i √ 2

Clearly, the above analysis shows that all that 1

is necessary to perform σz4 fault tolerantly is the 1

state |φ0 i and the ability to do ∧1 (σx ) and σz2 fault-tolerantly. For CSS codes, ∧1 (σx ), H and 1

σz2 can be done fault-tolerantly [25, 14]. We next show how to generate the state |φ0 i fault4 A Fault-Tolerant Realization tolerantly. 1 Fault tolerant creation of certain particular enof σz 4 coded eigenstates has been discussed[25, 18]. We present it in a more general way: suppose that We provide a simple scheme for the fault-tolerant 1 the fault-tolerant operation Uη operates as folrealization of the σz 4 gate. The method delows: scribes a general procedure that works for any Uη |ηi i = (−1)i |ηi i (16) quantum code for which the elements of the normalizer group can be implemented fault- on the states |η i. Thus, U has the states |η i as i η i tolerantly and involves the creation of special eigenvectors with ±1 as the eigenvalues. Suppose eigenstates of unitary transformations. we have access to a vector |ψi such that: 1

To perform σz4 fault-tolerantly we use the following state:

|ψi = α|η0 i + β|η1 i

(17)

We show that using only bitwise operations, π measurements, and this |ψi, the eigenvectors |ηi i 1 |0i + ei 4 |1i √ |φ0 i ≡ σz4 H|0i = , (15) can be obtained. Now, to get the eigenvector of 2 Uη we make use of a |cati state. |cati = √12 (|00...0i + |11...1i) = √12 (|~0i + |~1i) (for which we later present the preparation pro1 (18) cess). To apply σ 4 to a general single qubit z

state |ψi using this special state |φ0 i, first ap- See Figure 2. Applying ∧1 (Uη ) bitwise, on |cati⊗ ply ∧1 (σx ) from |ψi to |φ0 i. See Figure 1. Then |ψi we obtain: measure the second qubit (|φ0 i) in the computa∧1 (Uη ) ~ ~ ~ ~ 1 √ 1i )|η0 i + β( |0i−| √ 1i )|η1 i |cati ⊗ |ψi −→ α( |0i+| 2 2 2 tion basis. If the result is |1i, apply σz to the (19) first qubit (|ψi). This leads to the desired oper-

5

|~0i + |~1i

−1 σz σx σz 4 1 4

α|φ0 iL + β|φ1 iL

α(|~0i + |~1i)|φ0 iL + β(|~0i − |~1i)|φ1 iL

Figure 2: Creation of the |φ0 ieigenstate. Shor’s implementation of Toffoli [25] also uses a special case of this general procedure. For performing Toffoli one uses U = ∧1 (σz ) ⊗ σz to get the eigenstates:

A fault-tolerant measurement can be made to ~ ~ ~ ~ √ 1i from |0i−| √ 1i [25]. This meadistinguish |0i+| 2 2 surement can be repeated to verify that you have it correct. 1 The fault-tolerant version of σz4 needs the state |φ0 i, which can be generated using this formalism. In fact, |φ0 i is an eigenstate of 1

|ANDi =

−1

Uφ = σz4 σx σz 4 . By commutation properties of

1 (|000i + |010i + |100i + |111i) 2 1 (|001i + |011i + |101i + |110i) 2

|NANDi = −1 the σz 4 operator, it is shown that Uφ can be realized with elements only from the normalizer Shor uses the |ψi state of: Group. 1 √ (|ANDi + |NANDi) |ψi = 1 1 1 π − 2 Uφ = σz4 σx σz 4 = ei 4 σz2 σx . (20) = (H|0i) ⊗ (H|0i) ⊗ (H|0i) 1

Since σz2 and σx can be done fault-tolerantly, so Thus the special state in [25] can be obtained by can Uφ . We have claimed that |φ0 i (15) is an the same general procedure. eigenvector, and now we state the other eigenvector.

5

π

|0i − ei 4 |1i √ |φ1 i ≡ σz H|1i = 2 1 4

5.1

One can verify now that these |φi i are eigenvectors:

1 4

Uφ |φi i =

− 14

1

−1 σz σx σz 4 |φi i 1

1

= σz4 Hσz |ii = σz4 H(−1)i |ii = (−1)i |φi i Since the |φi i vectors are orthogonal, any single qubit state |ψi can be represented as a sum of the |φi i. |ψi = α|0i + β|1i = α′ |φ0 i + β ′ |φ1 i

(23)

So, all the necessary ingredients are here: |ψi, and an appropriate fault-tolerant operation, Uφ . If the outcome gives |φ1 i rather than |φ0 i we can flip the state: iπ 4

√ |φ0 i = σz |φ1 i = σz |0i−e 2

|1i

1

|0i+ei 4 |1i √ 2

1

Hz ≡ Hσz − 2 Hσz 2 H

∧2 (σz ) = (I4 ⊗ H) ∧2 (σx )(I4 ⊗ H)

π

=

1

Equivalence between {∧1 (σz 2 ), H} and Shor’s basis’

Kitaev ([17], Lemma 4.6) has provided a proof for the universality of the basis Q1 ≡ 1 This basis is equivalent to {∧1 (σz 2 ), H}. 1 Shor’s fault-tolerant basis {∧2 (σx ), σz 2 , H}, as observed in [17]. We give one demonstration of this equivalence here for completeness. This equivalence was also showed independently by Aharonov and Ben-Or (in journal version of [2]). To construct Q1 from Shor’s basis, it suffices 1 to construct ∧1 (σz 2 ). First we show that the operations ∧2 (σz ) and ∧2 (σy ) can be implemented exactly using gates from Shor’s basis (I2n is the identity operation on n qubits.):

1 4

= σz σx σz (σz4 H|ii) = σz4 σx H|ii (22) 1

Universality of Shor’s Basis

(21)

(24)

∧2 (σy ) = (I4 ⊗ Hz ) ∧2 (σx )(I4 ⊗ Hz ), 6

1

It will be shown that any gate in the set   0 0 0 1 X   0  ≡  0 SU(3)  0

Next, as shown in Figure 3, ∧1 (σz 2 ) can be implemented using the identity: σx σy σz = iI2 . This reduction also shows that the universality of Q1 implies the universality of Shor’s basis.

can be approximated to arbitrary precision by a two-qubit circuit consisting only of gates from the set G. I.e. the set G under regular matrix multiplication generates a set dense in Σ. From this set all single-qubit unitary operations SU(2) can be approximated, which along with ∧1 (σx ), has been shown [4] to be a universal set of gates. Though the correspondence is not a strict mathematical correspondence, it will be useful to make an analogy between real rotations in 3dimensional space, and gates constructible from G. Define the following 6 elements of hGi:

1

σz 2 σz

σy

σx

Figure 3: Constructing Q1 from Shor’s basis. Conversely, to construct Shor’s basis from Q1 , it suffices to construct the Toffoli gate ∧2 (σx ). 1 1 Note that ∧1 (σx ± 2 ) = (I2 ⊗ H) ∧1 (σz ± 2 )(I2 ⊗ H). Figure 4 gives the circuit construction of Toffoli (see Lemma 6.1 of [4] for a systematic construction of this circuit).

1

1

ρx ≡ ∧1(σx 2 ) ∧1 (σz 2 )−1 ρy ≡ Sρ−1 x S

ρz ≡ ∧1(σx )ρ−1 y ∧1 (σx )

1

σx 2

1

σx − 2

1

2 ρ1 ≡ ρ−1 z ∧1 (σx ) ∧1 (σz )ρz

1

σx 2

ρ2 ≡ ρx ρy

ρ3 ≡ ρ1 ρ2 ρ−1 1

Figure 4: Constructing ∧2 (σx ) from Q1 .

(Each inverse in the above definitions are obtainable from G, since each element of G is of finite 5.2 An Alternate Proof group order.) Since ρ2 and ρ3 are unitary, they An alternative proof, which makes use of irra- can be unitarily diagonalized: tional “rotations” about orthogonal axes, is preρ2 = g2 D(1, 1, e−i2πc , ei2πc )g2−1 sented in this section for the universality of each ρ3 = g3 D(1, 1, e−i2πc , ei2πc )g3−1 of the above two basis’. From either one of these sets, the following triplet of double-qubit gates where g2 , g3 are some unitary matrices (not necessarily in hGi), D is a diagonal matrix with the is constructible: given ordered quadruplet as√the entries along the 1 1 G ≡ {∧1 (σx 2 ), ∧1 (σz 2 ), S} diagonal, and ei2πc = 1+i4 15 for some c ∈ R. The minimum monic polynomial for ei2πc over where S is the swap gate: S|abi = |bai for any the set of rational numbers is single-qubit states |ai, |bi, which can be con1 structed as mei2πc (x) = x2 − x + 1 6∈ Z[x] 2 S = ∧1 (σx )(H ⊗ H) ∧1 (σx )(H ⊗ H) ∧1 (σx ). and thus c 6∈ Q (see Appendix B, Theorem B.1). It follows that successive powers of ρ2 and ρ3 can For future reference, note that each of the ma- approximate matrices of the forms trices in G are symmetric. Hence for any matrix ρn2 2 ≈ g2 D(1, 1, e−iθ2 , eiθ2 )g2−1 (25) M that is constructible from this set, so is its n −iθ iθ −1 transpose M T . ρ3 3 ≈ g3 D(1, 1, e 3 , e 3 )g3 (26) 7

for any θ2 , θ3 ∈ R. The powers n2 and n3 are grateful to D. Aharonov for her comments on an functions of θ2 and θ3 , as well as the desired de- earlier version of this work. gree of accuracy. The operators ρ1 , ρ2 , and ρ3 fix the (unnormalized) states |01i − |10i, |01i + |10i + |11i, and References − |01i − |10i + 2 |11i, resp. which form an or[1] L. M. Adleman, J. Demarrais, and M. -D. A. thogonal set of states. Motivated by considering Huang, “Quantum computability,” SIAM J. these 3 operations to be rotations about 3 orComputing, 26(1997), pp. 1524–1540. thogonal vectors, a change of basis is performed into this basis (while mapping the state |00i to it- [2] D. Aharonov and M. Ben-Or, “Faultself). Under this change of basis, equations (25) Tolerant Quantum Computation with Conand (26) are expressed as: stant Error,” Proc. of the 29th Annual   ACM Symposium on Theory of Computing 1 0 0 0 (STOC), pp. 46-55, 1997.   0 cos(θ2 ) 0 α sin(θ2 )  ρˆn2 2 ≈   0  0 1 0 [3] A. Barenco, “A universal two–bit gate for ∗ 0 −α sin(θ2 ) 0 cos(θ2 ) quantum computation,” Proc. Roy. Soc.   1 0 0 0 London Ser. A., 449(1995), pp. 679–683.  0 cos(θ3 ) −β ∗ sin(θ3 ) 0  n3  ρˆ3 ≈   0 β sin(θ3 ) [4] A. Barenco, C. H. Bennett, R. Cleve, D. cos(θ3 ) 0  P. DiVincenzo, N. Margolus, P. Shor, T. 0 0 0 1 Sleator, J. Smolin, H. Weinfurter, “Ele1+3i √ √ , which like ei2πc = and β ≡ where α ≡ 1+2i mentary Gates for Quantum Computation,” 5 10 √ 1+i 15 Phys. Rev. A, 52(1995), pp. 3457-3467. above, are also seen to not be roots of 4 unity. [5] E. Bernstein and U. Vazirani, “Quantum Given any γ ∈ C, |γ| = 1, define the following complexity theory,” SIAM J. Computing, single-parameter group of matrices: 26(1997), pp. 1411–1473.   cos(θ) −γ ∗ sin(θ) Mγ (θ) ≡ . [6] A. R. Calderbank and P. W. Shor, “Good γ sin(θ) cos(θ) quantum error–correcting codes exit,” Phys. If γ is not a root of unity, then it is straightRev. A, vol. 54, no. 2, pp. 1098–1105, 1996. forward to show that the set of matrices  Mγ (θ)T , Mγ (θ) θ ∈ R generates a dense sub- [7] J. I. Cirac and P. Zoller, “Quantum comset of SU(2). Given this, and the fact that putations with cold trapped ions”, Physiany element of SU(3) can be decomposed into cal Review Letters, Vol. 74, 15 May 1995, a product of SU(2) operations acting on orthogpp. 4091 – 4094. onal subspaces[21], it follows that the set  T [8] D. Deutsch, “Quantum theory, the Church– ρˆ2 , ρˆ2 , ρˆT3 , ρˆ3 Turing principle and the universal quantum computer,” Proc. Roy. Soc. London Ser. A., generates a dense subset of Σ. Since the previ400(1985), pp. 97–117. ous change of basis bijectively and continuously maps Σ onto itself, the operators G in the origi[9] D. Deutsch, ”Quantum computational netnal basis generates a dense subset of Σ.  works,” Proc. Roy. Soc. London Ser. A, 245(1989), pp. 73–90.

Acknowledgments

[10] P. Domokos, J.–M. Raimond, M. Brune and S. Haroche, “Simple cavity-QED two-bit universal quantum logic gate: The principle

We thank A. Kitaev and P. Shor for helpful discussions on the subject of this paper. We are

8

[11] [12]

[13]

[14]

[15]

[16]

[17]

and expected performances”, Physical Re- [22] J. Preskill, “Reliable quantum computers,” view A, November 1995, pp. 3554 – 3559. Proc. of the Royal Society of London, Ser. A, 454(1998), pp. 385–410. D. S. Dummit and R. M. Foote, Abstract [23] P. Shor, “Scheme for reducing decoherence Algebra, Prentice-Hall, Inc., 1991. in quantum computer memory,” Physical N. A. Gershenfeld and I. L. Chuang, “Bulk Review A, 52(1995), pp. 2493-6. spin-resonance quantum computation,” Sci[24] P. Shor, “Polynomial–time algorithms for ence, 275(1997), pp. 350–356. prime factorization and discrete logarithms D. G. Gory, A. F. Fahmy, and T. F. Havel, on a quantum computer,” SIAM J. Com“Ensemble quantum computing by nuclear puting, 26(1997), pp. 1484–1509. magnetic resonance spectroscopy,” in Proc. [25] P. W. Shor, “Fault-tolerant Quantum ComNatl. Acad. Sci. 94(1997), pp. 1634–1639. putation,” Proc. 37th Annual Symposium on Foundations of Computer Science, IEEE D. Gottesman, “Theory of fault-tolerant Computer Society Press, pp. 56-65, 1996. quantum computation,” Physical Review A, 57(1998), pp. 127-137. [26] J. J. Sakurai, Modern Quantum Mechanics, Revised Edition, Addison Wesley, 1994. L. Grover, “A fast quantum mechanical algorithm for database search,” in Proceed[27] R. Solovay and A. Yao, preprint, 1996. ings of 28th ACM Symposium on Theory of Computing, pp. 212–219, 1996. [28] A. M. Steane, “Error correcting codes in quantum theory,” Phys. Rev. Lett., vol. 77, B. E. Kane, “A silicon-based nuclear spin no. 5, pp. 793–797, 1996. quantum computer”, Nature, 393 (1998) pp. 133–137. [29] Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi and H. J. Kimble, “MeasureA. Kitaev, “Quantum Computations: Alment of conditional phase shifts for quangorithms and Error Correction,” Russian tum logic”, Phys. Rev. Lett., 75, pp. 4710– Math. Surveys 52(1997), pp. 1191-1249. 4713.

[18] E. Knill, R. Laflamme, and W. H. Zurek, [30] R. Vrijen, E. Yablonovitch, K. Wang, “Resilient quantum computation: error H. W. Jiang, A. Balandin, V. Roychowdmodels and thresholds,” Proceedings of hury, T. Mor, and D. DiVincenzo, “Electhe Royal Society of London, Series A, tron Spin Resonance Transistors for Quan454(1998), pp. 365-384. tum Computing in Silicon-Germanium Heterostructures”, Los-Alamos archive (1999) [19] E. Knill, R. Laflamme, and W. H. Zurek, quant-physics/9905096. “Accuracy threshold for quantum computation,” LANL eprint quant-ph/9611025.

A

Shor’s

basis

and

1 [20] C. Monroe, D. M. Meekhof, B. E. King, 4 , Λ (σ ) } are not { H, σ 1 x z W. M. Itano and D. J. Wineland, “Demonequivalent stration of a fundamental quantum logic gate”, Physical Review Letters, Vol. 75, In this appendix we show that Shor’s basis and 1995, pp. 4714–4717. 1 our basis { H, σz 4 , Λ1 (σx ) } are not equivalent. [21] F. D. Murnaghan, The Unitary and Rota- In fact, every gate in Shor’s basis can be exactly tion Groups, Spartan Books, Washington, represented by a circuit over our basis. First, the following identity shows that our basis can D.C., 1962.

9

exactly implement any gate from the Q1 basis • The minimum monic polynomial mα (x) ∈ introduced in Section 5.1: Q[x] for α ≡ ei2πc exists and is cyclotomic.   1 1 • c ∈ Q. I ⊗ σz − 4 Λ1 (σx ) Λ1 (σz 2 ) =   1 I ⊗ σz − 4 Λ1 (σx ) Proof:   1 A number of algebraic theorems will be taken for 1 σz 4 ⊗ σz 2 . granted in this proof, in particular, properties of cyclotomic polynomials Φn (x). See, for instance, Hence, as proved in the same section, it can ex- Dummit and Foote[11] for a more thorough disactly implement any gate from Shor’s basis. We cussion of these polynomials, as well as general prove that the converse is not true. Toward this properties of polynomial rings. 1 end, we show that the unitary operation σz 4 , Assume mα (x) exists and mα (x) = Φn (x) for can be computed exactly by our basis but not some n ∈ Z+ . by Shor’s basis. First we prove a useful Lemma about unitary operations computable exactly by 0 = 1 mα (α) Shor’s basis. Note that the set of integer com= 2 Φn (α) Y plex numbers is the set Z + iZ of the complex 3 Φd (α) = numbers with integer real and imaginary parts. 4

d|n n

Lemma A.1 Suppose that the unitary operation = α −1 5 U ∈ U(2m ) is the transformation performed by a = ei2πcn − 1. circuit C defined over Shor’s basis with m inputs. Then U is of the form √1 ℓ M , where M is a 2m × nc ∈ Z. Thus c ∈ Q. 2 Conversely, assume c ∈ Q. c = pq for some 2m matrix with only complex integer entries. p, q ∈ Z. mα (x) exists, since αq −1 = ei2πcq −1 = Proof. Suppose that g1 , . . . , gt are the gates ei2πp − 1 = 0. Moreover, mα (x) divides xq − 1 = Q of C. Each gate gj can be considered as a unid|q Φd (x) in Q[x]. mα (x) ∝ Φn (x) for some n|q. tary operation in U(2m ) by acting as an identity Since both are monic, mα (x) = Φn (x).  operator on the qubits that are not inputs of gj . Let the matrix Mj ∈ U(2m ) represent gj . Then 1 U = Mt · · · M1 . If gj is a σz 2 gate then Mj is a diagonal matrix with 1 or i on its diagonal. If gj is a Toffoli gate then Mj is a permutation matrix (which is a 0–1 matrix). Finally, if gj is a Hadamard gate, then Mj = √12 Mj ′ , where the entries of Mj ′ are integers. This completes the proof.  √  1 2 0 1 √ 4 Now since σz = 2 , by Lemma 0 1+i A.1 it cannot be realized exactly by gates from Shor’s basis.

B

The Cyclotomic/Rational Number Theorem

Theorem B.1 For any c ∈ R, the following two statements are logically equivalent:

10

1

Definition of mα (x) By assumption. 3 0 times anything is 0. 4 Property of cyclotomic polynomials. 5 Definition of α. 2